On $2$-generateness of weakly localizable submodules in the module of entire functions of exponential type and polynomial growth on the real axis

In the work we consider a topological module $\mathcal P(a;b)$ of entire functions, which is the isomorphic image of the Schwarz space of distributions with compact supports in a finite or infinite interval $(a;b)\subset\mathbb R$ under the Fourier–Laplace transform. We prove that each weakly localizable module in $\mathcal P (a;b)$ is either generated by its two elements or is equal to the closure of two submodules of special form. We also provide dual results on subspaces in $C^\infty(a;b)$ invariant w.r.t. the differentiation operator.